Algebra Examples

$\frac{2 a + b}{2 a^{2} - a b} - \frac{16 a}{4 a^{2} - b^{2}} - \frac{2 a - b}{2 a^{2} + a b}$

Step 1

Simplify each term.

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Step 1.1

Simplify the denominator.

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Step 1.1.1

Factor $a$ out of $2 a^{2} - a b$ .

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Step 1.1.1.1

Factor $a$ out of $2 a^{2}$ .

$\frac{2 a + b}{a (2 a) - a b} - \frac{16 a}{4 a^{2} - b^{2}} - \frac{2 a - b}{2 a^{2} + a b}$

Step 1.1.1.2

Factor $a$ out of $- a b$ .

$\frac{2 a + b}{a (2 a) + a (- 1 b)} - \frac{16 a}{4 a^{2} - b^{2}} - \frac{2 a - b}{2 a^{2} + a b}$

Step 1.1.1.3

Factor $a$ out of $a (2 a) + a (- 1 b)$ .

$\frac{2 a + b}{a (2 a - 1 b)} - \frac{16 a}{4 a^{2} - b^{2}} - \frac{2 a - b}{2 a^{2} + a b}$

Step 1.1.2

Rewrite $- 1 b$ as $- b$ .

$\frac{2 a + b}{a (2 a - b)} - \frac{16 a}{4 a^{2} - b^{2}} - \frac{2 a - b}{2 a^{2} + a b}$

Step 1.2

Simplify the denominator.

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Step 1.2.1

Rewrite $4 a^{2}$ as ${(2 a)}^{2}$ .

$\frac{2 a + b}{a (2 a - b)} - \frac{16 a}{{(2 a)}^{2} - b^{2}} - \frac{2 a - b}{2 a^{2} + a b}$

Step 1.2.2

Since both terms are perfect squares, factor using the difference of squares formula, $a^{2} - b^{2} = (a + b) (a - b)$ where $a = 2 a$ and $b = b$ .

$\frac{2 a + b}{a (2 a - b)} - \frac{16 a}{(2 a + b) (2 a - b)} - \frac{2 a - b}{2 a^{2} + a b}$

Step 1.3

Factor $a$ out of $2 a^{2} + a b$ .

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Step 1.3.1

Factor $a$ out of $2 a^{2}$ .

$\frac{2 a + b}{a (2 a - b)} - \frac{16 a}{(2 a + b) (2 a - b)} - \frac{2 a - b}{a (2 a) + a b}$

Step 1.3.2

Factor $a$ out of $a b$ .

$\frac{2 a + b}{a (2 a - b)} - \frac{16 a}{(2 a + b) (2 a - b)} - \frac{2 a - b}{a (2 a) + a (b)}$

Step 1.3.3

Factor $a$ out of $a (2 a) + a (b)$ .

$\frac{2 a + b}{a (2 a - b)} - \frac{16 a}{(2 a + b) (2 a - b)} - \frac{2 a - b}{a (2 a + b)}$

Step 2

To write $\frac{2 a + b}{a (2 a - b)}$ as a fraction with a common denominator, multiply by $\frac{2 a + b}{2 a + b}$ .

$\frac{2 a + b}{a (2 a - b)} \cdot \frac{2 a + b}{2 a + b} - \frac{16 a}{(2 a + b) (2 a - b)} - \frac{2 a - b}{a (2 a + b)}$

Step 3

To write $- \frac{16 a}{(2 a + b) (2 a - b)}$ as a fraction with a common denominator, multiply by $\frac{a}{a}$ .

$\frac{2 a + b}{a (2 a - b)} \cdot \frac{2 a + b}{2 a + b} - \frac{16 a}{(2 a + b) (2 a - b)} \cdot \frac{a}{a} - \frac{2 a - b}{a (2 a + b)}$

Step 4

Write each expression with a common denominator of $a (2 a - b) (2 a + b)$ , by multiplying each by an appropriate factor of $1$ .

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Step 4.1

Multiply $\frac{2 a + b}{a (2 a - b)}$ by $\frac{2 a + b}{2 a + b}$ .

$\frac{(2 a + b) (2 a + b)}{a (2 a - b) (2 a + b)} - \frac{16 a}{(2 a + b) (2 a - b)} \cdot \frac{a}{a} - \frac{2 a - b}{a (2 a + b)}$

Step 4.2

Multiply $\frac{16 a}{(2 a + b) (2 a - b)}$ by $\frac{a}{a}$ .

$\frac{(2 a + b) (2 a + b)}{a (2 a - b) (2 a + b)} - \frac{16 a \cdot a}{(2 a + b) (2 a - b) a} - \frac{2 a - b}{a (2 a + b)}$

Step 4.3

Reorder the factors of $a (2 a - b) (2 a + b)$ .

$\frac{(2 a + b) (2 a + b)}{a (2 a + b) (2 a - b)} - \frac{16 a \cdot a}{(2 a + b) (2 a - b) a} - \frac{2 a - b}{a (2 a + b)}$

Step 4.4

Reorder the factors of $(2 a + b) (2 a - b) a$ .

$\frac{(2 a + b) (2 a + b)}{a (2 a + b) (2 a - b)} - \frac{16 a \cdot a}{a (2 a + b) (2 a - b)} - \frac{2 a - b}{a (2 a + b)}$

Step 5

Combine the numerators over the common denominator.

$\frac{(2 a + b) (2 a + b) - 16 a \cdot a}{a (2 a + b) (2 a - b)} - \frac{2 a - b}{a (2 a + b)}$

Step 6

Simplify each term.

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Step 6.1

Simplify the numerator.

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Step 6.1.1

Raise $2 a + b$ to the power of $1$ .

$\frac{{(2 a + b)}^{1} (2 a + b) - 16 a \cdot a}{a (2 a + b) (2 a - b)} - \frac{2 a - b}{a (2 a + b)}$

Step 6.1.2

Raise $2 a + b$ to the power of $1$ .

$\frac{{(2 a + b)}^{1} {(2 a + b)}^{1} - 16 a \cdot a}{a (2 a + b) (2 a - b)} - \frac{2 a - b}{a (2 a + b)}$

Step 6.1.3

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$\frac{{(2 a + b)}^{1 + 1} - 16 a \cdot a}{a (2 a + b) (2 a - b)} - \frac{2 a - b}{a (2 a + b)}$

Step 6.1.4

Add $1$ and $1$ .

$\frac{{(2 a + b)}^{2} - 16 a \cdot a}{a (2 a + b) (2 a - b)} - \frac{2 a - b}{a (2 a + b)}$

Step 6.1.5

Rewrite $16 a \cdot a$ as ${(4 a)}^{2}$ .

$\frac{{(2 a + b)}^{2} - {(4 a)}^{2}}{a (2 a + b) (2 a - b)} - \frac{2 a - b}{a (2 a + b)}$

Step 6.1.6

Since both terms are perfect squares, factor using the difference of squares formula, $a^{2} - b^{2} = (a + b) (a - b)$ where $a = 2 a + b$ and $b = 4 a$ .

$\frac{(2 a + b + 4 a) (2 a + b - (4 a))}{a (2 a + b) (2 a - b)} - \frac{2 a - b}{a (2 a + b)}$

Step 6.1.7

Simplify.

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Step 6.1.7.1

Add $2 a$ and $4 a$ .

$\frac{(6 a + b) (2 a + b - (4 a))}{a (2 a + b) (2 a - b)} - \frac{2 a - b}{a (2 a + b)}$

Step 6.1.7.2

Multiply $4$ by $- 1$ .

$\frac{(6 a + b) (2 a + b - 4 a)}{a (2 a + b) (2 a - b)} - \frac{2 a - b}{a (2 a + b)}$

Step 6.1.7.3

Subtract $4 a$ from $2 a$ .

$\frac{(6 a + b) (- 2 a + b)}{a (2 a + b) (2 a - b)} - \frac{2 a - b}{a (2 a + b)}$

Step 6.2

Cancel the common factor of $- 2 a + b$ and $2 a - b$ .

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Step 6.2.1

Factor $- 1$ out of $- 2 a$ .

$\frac{(6 a + b) (- (2 a) + b)}{a (2 a + b) (2 a - b)} - \frac{2 a - b}{a (2 a + b)}$

Step 6.2.2

Factor $- 1$ out of $b$ .

$\frac{(6 a + b) (- (2 a) - 1 (- b))}{a (2 a + b) (2 a - b)} - \frac{2 a - b}{a (2 a + b)}$

Step 6.2.3

Factor $- 1$ out of $- (2 a) - 1 (- b)$ .

$\frac{(6 a + b) (- (2 a - b))}{a (2 a + b) (2 a - b)} - \frac{2 a - b}{a (2 a + b)}$

Step 6.2.4

Rewrite $- (2 a - b)$ as $- 1 (2 a - b)$ .

$\frac{(6 a + b) (- 1 (2 a - b))}{a (2 a + b) (2 a - b)} - \frac{2 a - b}{a (2 a + b)}$

Step 6.2.5

Cancel the common factor.

$\frac{(6 a + b) (- 1 (2 a - b))}{a (2 a + b) (2 a - b)} - \frac{2 a - b}{a (2 a + b)}$

Step 6.2.6

Rewrite the expression.

$\frac{(6 a + b) \cdot (- 1)}{a (2 a + b)} - \frac{2 a - b}{a (2 a + b)}$

Step 6.3

Move $- 1$ to the left of $6 a + b$ .

$\frac{- 1 \cdot (6 a + b)}{a (2 a + b)} - \frac{2 a - b}{a (2 a + b)}$

Step 6.4

Move the negative in front of the fraction.

$- \frac{6 a + b}{a (2 a + b)} - \frac{2 a - b}{a (2 a + b)}$

Step 7

Combine the numerators over the common denominator.

$\frac{- (6 a + b) - (2 a - b)}{a (2 a + b)}$

Step 8

Simplify the numerator.

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Step 8.1

Simplify each term.

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Step 8.1.1

Apply the distributive property.

$\frac{- (6 a) - b - (2 a - b)}{a (2 a + b)}$

Step 8.1.2

Multiply $6$ by $- 1$ .

$\frac{- 6 a - b - (2 a - b)}{a (2 a + b)}$

Step 8.1.3

Apply the distributive property.

$\frac{- 6 a - b - (2 a) - - b}{a (2 a + b)}$

Step 8.1.4

Multiply $2$ by $- 1$ .

$\frac{- 6 a - b - 2 a - - b}{a (2 a + b)}$

Step 8.1.5

Multiply $- - b$ .

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Step 8.1.5.1

Multiply $- 1$ by $- 1$ .

$\frac{- 6 a - b - 2 a + 1 b}{a (2 a + b)}$

Step 8.1.5.2

Multiply $b$ by $1$ .

$\frac{- 6 a - b - 2 a + b}{a (2 a + b)}$

Step 8.2

Combine the opposite terms in $- 6 a - b - 2 a + b$ .

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Step 8.2.1

Add $- b$ and $b$ .

$\frac{- 6 a - 2 a + 0}{a (2 a + b)}$

Step 8.2.2

Add $- 6 a - 2 a$ and $0$ .

$\frac{- 6 a - 2 a}{a (2 a + b)}$

Step 8.3

Subtract $2 a$ from $- 6 a$ .

$\frac{- 8 a}{a (2 a + b)}$

Step 9

Cancel the common factor of $a$ .

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Step 9.1

Cancel the common factor.

$\frac{- 8 a}{a (2 a + b)}$

Step 9.2

Rewrite the expression.

$\frac{- 8}{2 a + b}$

Step 10

Move the negative in front of the fraction.

$- \frac{8}{2 a + b}$